It is well-known that for $n \geq 2$ and a finite field $k$, the Milnor $K$-group $K_n ^M (k)$ vanishes. I don't know who proved this first, but if curious, you may look at somewhere in Srinivas's book "Algebraic K-theory" or various lecture notes in the internet.
My question is whether it is always nonzero, when $k$ is infinite. Certainly we have concrete computations in the literature where they are nonzero, and results such as Nesterenko-Suslin-Totaro, which shows Milnor K-groups are higher Chow groups of $k$ in the Milnor range. But, they don't seem to imply anything about non-vanishing of $K_n ^M (k)$ for $n \geq 2$.
It is also hard to prove that some group is nonzero even though you know the generators and relations. (In a sense, if true, one "could" possibly prove something is zero by computations, but something is "nonzero" seems to require something else.
Does anyone have some ideas about it?